Are AI algorithms capable of self-repair?
Yes, this was an active area of research in a number of different AI fields.

Probably the most directly related work is Bongard, Zykov &amp; Lipson's self-repairing robots from the early 2000's. 

There's some more recent work from Mark Yim that you can see here too.

There are lots of different ways to do this, but Bongard et al's approach was probably the most elegant. The basic idea was to frame it as a learning problem: the robot is able to learn the shape of its body by performing controlled experiments. When the body is damaged, the robot can detect that it's body has changed shape (sensors don't report the expected values when it tries to move), perform new experiments to determine the extent of the damage, and then generate new movements that work around the damaged area. Lipson covers the basics of this system very briefly in this video.

The more modern system uses a similar approach, but tries to repair its body, rather than working around the damage. It's got an internal model of what it's body should look like, and then a set of cameras that help it locate the various pieces and move them to reassemble. 

Dropout is sort of a similar idea, but dropout is usually done to encourage redundancy during training, which can help a model avoid overfitting. It's usually not done explicitly to heal a damaged system, although it would make a system more resistant to damage in the first place.